= Special linear congruence action on symmetric matrices
{title2=$A\cdot B=ABA^T,\quad\det A=\det B=1$}
The <special linear group> acts on determinant-one real <symmetric matrices> by <matrix congruence>. Symmetry and <determinant> are preserved, and signatures are preserved by <Sylvester's law of inertia>. The determinant-one set is a <regular level set> of dimension $n(n+1)/2-1$, with tangent directions $C=C^T$ satisfying $\operatorname{tr}(B^{-1}C)=0$. Its infinitesimal left-action fields are $XB+BX^T$; converting them to a <Lie algebra representation> requires the <infinitesimal left-action sign convention>.
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